How can you show that the complete elliptic integral of first kind
$
\displaystyle K(m)=\int_0^\frac{\pi}{2}\frac{\mathrm du}{\sqrt{1-m^2\sin^2 u}}$
that is the same as a series
$$K(m)=\frac{\pi}{2} \left(1+\left(\frac{1}{2}\right)^{2}m^2 +\left(\frac{1\cdot 3}{2\cdot 4}\right)^{2}m^4 +...+ \left(\frac{(2n-1)!!}{2n!!} \right )^2m^{2n} + ... \right)$$

Yes, something was definitely wrong with the expansion... But it seems you saw the problem since you made the necessary correction. Note that this makes the accepted answer, which addresses (incorrectly) the original version of your question, a little odd.
–
DidDec 8 '12 at 23:42